A111723 Number of partitions of an n-set with an odd number of blocks of size 1.

1, 0, 4, 4, 31, 86, 449, 1968, 10420, 56582, 333235, 2069772, 13606113, 94065232, 682242552, 5175100432, 40954340995, 337362555010, 2886922399649, 25616738519384, 235313456176512, 2234350827008170, 21899832049913999, 221292603495494488, 2302631998398438321

Offset: 1

Author

Vladeta Jovovic, Nov 17 2005

Keywords

• easy
• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..576

Crossrefs

Cf. A097514, A113235, A063083, A062282, A111724, A111752, A111753.

Programs

Maple

b:= proc(n, t) option remember; `if`(n=0, t, add(b(n-j,
      `if`(j=1, 1-t, t))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=1..30);  # Alois P. Heinz, May 10 2016

Mathematica

Rest[ Range[0, 23]! CoefficientList[ Series[ Sinh[x]Exp[Exp[x] - 1 - x], {x, 0, 23}], x]] (* Robert G. Wilson v *)

Python

from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def b(n, t):
    return t if n==0 else sum(b(n - j, (1 - t if j==1 else t))*binomial(n - 1, j - 1) for j in range(1, n + 1))
def a(n):
    return b(n, 0)
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Aug 10 2017

Formula

E.g.f.: sinh(x)*exp(exp(x)-1-x).

More generally, e.g.f. for number of partitions of an n-set with an odd number of blocks of size k is sinh(x^k/k!)*exp(exp(x)-1-x^k/k!).

Extensions

More terms from Robert G. Wilson v, Nov 22 2005